I have one scenario in which case I can't get the data of a push notification into my app.
When I send one out and my app is closed, I'd like the data in the app when I open my app again (for instance after 5 minutes or so).

Is this possible? And if so, how can this be achieved with the Cordova library?

Update is here. The plugin 3.4.2 is out.
It features experimental getPushHistory() / clearPushHistory() methods for Android only (!!!)
By calling this methods you can receive a JSONArray of pushes that has arrived to the app. Only last 10 pushes being kept. Don't forget to call clearPushHistory as you fetch the array.